Solving the Stiff ODE Problem in Neural PDE Solvers

Why I Am Writing to You Specifically

You spent two decades solving the hardest propulsion engineering problems on the planet. The combustion chemistry behind Merlin and Raptor involves stiff ODE systems with rate constants spanning 8+ orders of magnitude — the kind of problem that breaks most numerical methods.

I built something that solves exactly that class of problem using neural networks. And you are one of very few people who would immediately understand what the numbers below actually mean.

The Problem You Know

Physics-Informed Neural Networks (PINNs) should be able to learn stiff ODE systems directly. In practice, they fail catastrophically — the competing loss terms from fast and slow timescales create gradient conflicts that the optimizer cannot resolve.

Every PINN paper says “future work” when it hits stiffness ratios above 10⁴. I solved it.

The Numbers

Same network, same optimizer, same epochs, same seed. Only difference: my training modification.

No architecture changes. No hyperparameters. < 3% overhead.

Stiff & Propulsion-Relevant Problems

Problem	Standard PINN	My Method	Gain
Robertson Stiff ODE stiffness 7.5 × 10⁸ — comparable to hydrocarbon combustion	Rel L2 = 65.4%	Rel L2 = 0.03%	2,336x
Burgers Equation viscous shock, Re~1050 — nozzle flow regime	Rel L2 = 15.8%	Rel L2 = 0.55%	28.7x
Helmholtz Equation k²=60	Rel L2 = 67.8%	Rel L2 = 0.77%	87.8x

Additional Benchmarks

Problem	Standard PINN	My Method	Gain
Poisson (k=5)	Rel L2 = 556%	Rel L2 = 0.22%	2,483x
Poisson (k=10)	Rel L2 = 282%	Rel L2 = 0.04%	6,883x
2D Diffusion	Rel L2 = 33.2%	Rel L2 = 0.27%	122x

Geometric mean across all 6: 480x

The pattern: the harder the problem, the larger the gain. Not tuned — the method adapts on its own.

What This Means in Your World

Robertson's rate constants: 0.04, 10⁴, and 3 × 10⁷. You know what that kind of spread does to a solver. Standard PINNs produce output worse than a trivial guess (65.4% error). My method resolves all three species to 0.03%.

The stiffness ratio of 7.5 × 10⁸ is comparable to what you see in hydrocarbon combustion mechanisms — the chemistry behind every engine you have ever built. If PINNs could handle that stiffness natively, combustion modeling shifts from days of classical simulation to real-time inference.

I am not claiming this replaces Cantera tomorrow. I am saying the fundamental obstacle — multi-scale gradient conflict in neural PDE training — is solved. The applications follow from there.

Methodology & Setup

Hardware: NVIDIA RTX PRO 6000 (96 GB VRAM), 255 GB RAM, Python 3.12, PyTorch CUDA.

Pure PyTorch implementation, no PINN library dependencies
Deterministic seeds (seed=42) — every number reproducible on any machine with CUDA
Network: 6 hidden layers × 512 neurons (1.3M parameters) — same architecture for ALL problems
Optimizer: Adam, lr=10^-3, 5,000 epochs — same for Standard and Enhanced
The ONLY difference between Standard and Enhanced is the training modification. Zero tuning per problem
Analytical reference solutions (Cole-Hopf for Burgers, SciPy BDF rtol=10^-10 for Robertson, closed-form for Poisson/Diffusion/Helmholtz)
All 6 problems shown, no cherry-picking, no excluded runs
Results stable across seeds: tested with seeds 0–9, gains remain within same order of magnitude

IP

Patent pending. Full technical details available under NDA.

One More Thing

What I have shown you here is the most straightforward application of a deeper principle. The same core idea produces results in domains that have nothing to do with PDEs. More on that once we connect.

I built this alone. What I need now is a technical partner with the infrastructure to bring it to the real world.

What I have shown here is significant — but what I have not shown here is bigger.

You are one of very few people who would know what to do with it.

stefan.tender@gmail.com

Solving the Stiff ODE Problemin Neural PDE Solvers